Definition: Let F be a subset of a metric space X. F is called(adsbygoogle = window.adsbygoogle || []).push({}); closedif whenever is a sequence in F which converges to a E X, then a E F. (i.e. F contains all limits of sequences in F) Theclosureof F is the set of all limits of sequences in F.

Claim 1: F is contained in the clousre of F.

Claim 2: The closure of F is closed.

How can we prove these formally?

For claim 1, I think we have to show x E F => x E cl(F).

For claim 2, we need to prove that cl(F) contains all limits of sequences in cl(F).

But how can we prove these?

I know these are supposed to be basic facts, but my book never gives examples of how to prove these from the definitions given above...and I have no clue...

Any help is appreciated!

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# Closure & Closed Sets in metric space

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